We are used to thinking about space as a smooth continuum, which can, in principle, at least, be probed to infinitely small dimensions. For instance, in Euclidean geometry, a point is defined as "that which has no part." In other words, points have no volume, area, or length, and yet they are fundamental objects in Euclidean geometry. (Figure 1 shows part of one of the oldest surviving copies of Euclid's The Elements.)
Figure 1. One of the oldest surviving copies of a part of Euclid's book on the elements of geometry, from circa 100 CE (image in the public domain).
It may therefore come as a surprise that in modern physics, there exists a scale below which the very notions of length and space cease to exist. This is known as the Planck length, named after the German physicist Max Planck (Figure 2).
Figure 2. The German theoretical physicist Max Planck (1858-1947) (image in the public domain)
The Planck length is defined using three so-called "constants of nature": Newton's gravitational constant G (which determines the strength of gravity), the speed of light c, and the constant characterizing all subatomic quantum phenomena ħ (pronounced "h-bar"). The Planck length is given by:
To get an approximate appreciation for the value of this tiny length, consider the following. With the unaided human eye we can see dots that are a bit smaller than one-tenth of a millimeter in size. Such dots are larger than the Planck scale by the same factor that the entire observable universe is larger than the dots themselves.
You may think: If the Planck scale is so tiny, why should we even discuss it? The reason is that in attempts to unify the theory of gravity -- general relativity -- with the theory of the subatomic world and light -- quantum mechanics -- the Planck length may be the shortest measurable length. That is, irrespective of how much we would improve our measurement instruments, we would not be able to measure shorter dimensions. Since the way we probe tiny distances in the subatomic regime is through high-energy collisions, this limitation translated into the statement that no matter what energies we would use, we would not split spacetime into finer pieces. In fact, while no precise prediction can be made in quantum gravity (since no self-consistent theory has been formulated so far), some physicists guess that spacetime may behave like a discrete type of "foam" at the Planck scale. In string theory, still the leading candidate for a theory of all the fundamental particles and forces, the Planck length represents the size of the oscillating loops whose vibrations form the elementary particles. To paraphrase William Blake, who wrote, "To see a world in a grain of sand," we can perhaps say, "To see a world in a grain of Planck length."